The Proof Spine is the mathematical core of the Atlas. Where other books describe what systems should do, this book proves what they are allowed to do. It defines admissible action as a formal object and shows how safety, consent, reversibility, resilience, and human agency can be encoded as mathematical constraints rather than ethical suggestions. Instead of optimizing outcomes and hoping for the best, this book inverts the logic of power. Harm is treated as a forbidden region. Consent becomes a logical gate. Exit becomes a reachability condition. Uncertainty shrinks the space of allowed action. Emergency tightens constraints instead of loosening them. Null action is always preserved as a valid choice. Book 15 provides the shared proof layer used by every other volume in the Atlas. Policies, machines, institutions, and algorithms must pass through this spine before they are permitted to act on people or resources. Optimization is only allowed inside admissible sets; outside them, action collapses to silence. This book does not tell the world what to do. It specifies what must never be done unless formally justified. By translating dignity, restraint, and reversibility into barrier functions, invariant sets, and reachability proofs, it makes unjustifiable action structurally impossible rather than merely discouraged. Book 15 sits beneath all other layers. It answers one question with mathematical precision: When is power allowed to appear at all? Its answer is deliberately conservative and rigorously provable: Only after its limits are demonstrated, its exits are guaranteed, and its harms are excluded by construction.